Think about statistical inferences from the ground up again

The evidence in large medical data sets is direct, but indirect as well – and there is just too much of the indirect evidence to ignore. If you want to prove that your drug of choice is good or bad your evidence is not just how it does, it is also how all the other drugs do. And that is a crucial point that doesn’t fit easily into the frequentist world, which is a world of direct evidence (very often, but not always); and it also doesn’t fit extremely well into the formal Bayesian world, because the indirect information isn’t actually the prior distribution, it is evidence of a prior distribution, and that in some sense is not as neat. Neatness counts in science. Things that people can understand and really manipulate are terribly important.

“So I have been very interested in massive data sets not because they are massive but because they seem to offer opportunities to think about statistical inferences from the ground up again.”

The Fisher–Pearson –Neyman paradigm dating from around 1900 was, he says, “like a light being switched on. But it is so beautiful and so almost airtight that it is pretty hard to improve on; and that means that it is very hard to rethink what is good or bad about
statistics.

“Fisher of course had this wonderful view of how you do what I would call small-sample inference. You tend to get very smart people trying to improve on this kind of area, but you really cannot do that very well because there is a limited amount that is available to work on. But now suddenly there are these problems that have a different flavour. It really is quite different doing ten thousand estimates at once. There is evidence always lurking around the edges. It is hard to say where that evidence is, but it’s there. And if you ignore it you are just not going to do a good job.

“Another way to say it is that a Bayesian prior is an assumption of an infinite amount of past relevant experience. It is an incredibly powerful assumption, and often a very useful assumption for moving forward with complicated data analysis. But you cannot forget that you have just made up a whole bunch of data.

“So of course the trick for Bayesians is to do their ‘making up’ part without really influencing the answer too much. And that is really
tricky in these higher-dimensional problems.”

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Efron always has interesting things to say, but I think it is a mistake to suppose that Fisher-Neymen-Pearson theory is some closed system. Neyman was clearly excited by the newer work combining studies to improve overall error probability constraints, not very different from the lines of Efron’s “indirect” inference and empirical Bayes. Although Egon Pearson didn’t gravitate so much toward such behavioral developments, he to emphasized*.
“There is perhaps in current literature a tendency to speak of the Neyman Pearson contributions as some static system, rather than as part of the historical process of development of thought on statistical theory which is and will always go on” (Pearson 1966, 276). (*chapter 9, p. 317 of Mayo 1996).http://www.phil.vt.edu/dmayo/personal_website/EGEK CH NINE.pdf